Question

What is ln x + 2 ln y - ln z written as one logarithm?

Answer 1

Hello there. To solve this question, we have to remember some properties about logarithms.

Given the expression:

`\ln(x)+2\ln(y)-\ln(z)`

We want to rewrite it as a single logarithm.

For this, remember the following rules:

`\begin{gathered} \log_a(b)+\log_a(c)=\log_a(b\cdot c) \\ \\ \log_a(b)-\log_a(c)=\log_a\left((b)/(c)\right) \\ \\ c\cdot\log_a(b)=\log_a(b^c) \end{gathered}`

In this case we apply the third rule to the middle logarithm in order to get:

`\ln(x)+\ln(y^2)-\ln(z)`

Apply the first and second rules

`\begin{gathered} \ln(xy^2)-\ln(z)\text{ \lparen First rule\rparen} \\ \\ \Rightarrow\ln\left((xy^2)/(z)\right)\text{ \lparen Second rule\rparen} \end{gathered}`

This is the answer to this question and it is contained in the first option.